\contentsline {section}{\numberline {1}Introduction}{3}{section.1}%
\contentsline {section}{\numberline {2}Project Overview}{3}{section.2}%
\contentsline {section}{\numberline {3}Implementation Summary}{4}{section.3}%
\contentsline {subsection}{\numberline {3.1}Domain Decomposition}{4}{subsection.3.1}%
\contentsline {subsection}{\numberline {3.2}Neural Network Architecture}{4}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}Continuity Enforcement}{4}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}Solver Stack}{5}{subsection.3.4}%
\contentsline {subsection}{\numberline {3.5}Utilities and Examples}{5}{subsection.3.5}%
\contentsline {section}{\numberline {4}Experimental Evaluation}{5}{section.4}%
\contentsline {subsection}{\numberline {4.1}One-Dimensional Helmholtz Equation}{5}{subsection.4.1}%
\contentsline {subsection}{\numberline {4.2}Two-Dimensional Helmholtz Equation}{6}{subsection.4.2}%
\contentsline {subsection}{\numberline {4.3}Advection Equation Benchmarks}{7}{subsection.4.3}%
\contentsline {subsection}{\numberline {4.4}Diffusion and Nonlinear Helmholtz Studies}{7}{subsection.4.4}%
\contentsline {subsection}{\numberline {4.5}Burgers Equation Experiments}{8}{subsection.4.5}%
\contentsline {section}{\numberline {5}Analysis and Implementation Review}{8}{section.5}%
\contentsline {subsection}{\numberline {5.1}Interpretation of Experimental Results}{9}{subsection.5.1}%
\contentsline {subsection}{\numberline {5.2}Implementation Assessment}{9}{subsection.5.2}%
\contentsline {subsection}{\numberline {5.3}Targeted Improvement Opportunities}{10}{subsection.5.3}%
\contentsline {section}{\numberline {6}Outstanding Issues}{11}{section.6}%
\contentsline {subsection}{\numberline {6.1}Accuracy Discrepancy in 1D}{11}{subsection.6.1}%
\contentsline {subsection}{\numberline {6.2}Performance Bottleneck in 2D}{12}{subsection.6.2}%
\contentsline {section}{\numberline {7}Recommendations}{12}{section.7}%
\contentsline {subsection}{\numberline {7.1}Critical Fixes}{12}{subsection.7.1}%
\contentsline {subsection}{\numberline {7.2}Validation Enhancements}{12}{subsection.7.2}%
\contentsline {subsection}{\numberline {7.3}Extensions and Optimisations}{12}{subsection.7.3}%
\contentsline {section}{\numberline {8}Conclusion}{12}{section.8}%
